Let me give you some context first: just a few days ago I found some intriguing references to Ricci flows in the setting of directed graphs.

There are at least two versions of **Ricci curvature** in the discrete realm (one being the **Ollivier-Ricci** curvature, the other the **Forman-Ricci**, see here for reference (*)), and as it turns out, they are both useful in graph analytics.

To be a tad more specific, one application leads to a new method for determining communities (the so-called **Ricci communities**, for the interested ones there is even a github Python implementation which can easily be used for hands-on explorations ), whereas another quite useful one is used to get rid of "bottlenecks" in graph messaging ( thereby solving some critical issue in **Graph deep learning** see picture below).

https://towardsdatascience.com/over-squashing-bottlenecks-and-graph-ricci-curvature-c238b7169e16

Now, if I understand them correctly, the associated Ricci flow, just like in the differentiable realm, acts as a kind of "curvature heat-like operator", a diffusion which tends to smoothen out the curvature across the underlying geometrical object.

Perhaps naively, it occurred to me this:

*why confining ourselves to diffusion*? (note: I am aware of the centrality of the Ricci flow in the proof of the Poincare conjecture)

Could one replace the Ricci flow with some kind of PDE (or a difference equation in the finite setting) for the curvature change modeled on completely different PDEs?

For instance, what about a kind of wave equation?

Now the questions (and I apologize if this is too naive, I am coming from the data science world, my knowledge of Riemannian geometry does not go beyond standard grad courses):

- Have such curvature flow involving non-heat-like PDEs been investigated in the world of Riemannian geometry? I would think the answer is in the affirmative, but I just do not happen to know it.
- Are there any references for generalized curvature flows in discrete metric spaces and particularly in weighted directed graphs?

Any help is most welcome.

(*) actually in the referenced article there are three discrete Ricci curvatures, but I haven't wrapped my mind around the third one yet.

natural form. We hope thatnatural formto provide us with some geometric picture thatunderstandsthe topology best, abest metric. But with a wave equation, or a Schrodinger equation, or a Yang-Mills equation, we expect it to just keep bouncing around. So there is no approaching abest metric. So the first question seems to be what we would hope for in a wave equation. $\endgroup$